A separable Fréchet space of almost universal disposition

Abstract

The Gurari\uı space is the unique separable Banach space $\mathbb{G}$ which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every $\epsilon >0$, for all finite-dimensional normed spaces $E\subseteq F$, for every isometric embedding $e:E\to \mathbb{G}$ there exists an $\epsilon$-isometric embedding $f:F\to \mathbb{G}$ such that $f\upharpoonright E=e$. We show that ${\mathbb{G}}^{\mathbb{N}}$ with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fréchet spaces. The construction relies heavily on the universal operator on the Gurari\uı space, recently constructed by Garbulińska-Wegrzyn and the third author. This yields in particular that ${\mathbb{G}}^{\mathbb{N}}$ is universal in the class of all separable Fréchet spaces.

15 pages

References in corpus (2)

• Banach spaces of universal disposition
• Quasi-Banach spaces of almost universal disposition

Cited by in corpus (1)

• A note on universal operators between separable Banach spaces